Demysifying Latschev’s Theorem

Vietoris–Rips complexes for manifold reconstruction

Sush Majhi, George Washington University

Today’s Agenda

  • The problem of shape reconstruction
  • Different reconstruction paradigms
  • Vietoris–Rips complexes
  • Theorems of Hausmann and Latschev
  • Demistifying Lastchev’s theorem
    • Jung’s theorem on manifolds
  • Euclidean submanifold reconstruction
  • Question

The Problem of Shape Reconstruction

  • Shape: A Shape is modeled as a metric space \((M,d_M)\).

    • general compact set

    • metric graph

    • for today, \(M\) is a closed Riemannian manifold.

  • Sample: A finite metric space \((X,d_X)\) close to \(M\).

    • small Hausdorff proximity if \(M\) is a Euclidean submanifold and \(X\subset\mathbb R^d\)

    • for today, small Gromov–Hausdorff distance

  • Goal: Infer the topology of \(M\) from \(X\).

    • Estimate only the betti numbers: number of connected components, cycles, voids, etc, of \(M\).

    • for today, construct a topological space \(\widetilde{M}\) from \(X\) to retain the topology of \(M\), i.e., \(M\simeq\widetilde{M}\).

Vietoris–Rips Complex

  • a metric space \((X,d_X)\)

  • a scale \(\beta>0\),

  • \(\mathcal{R}_\beta(X)\) is an abstract simplicial complex such that

    • each subset \(A\subset X\) of size \(k\) with diameter at most \(\epsilon\) is a \((k-1)\)-simplex.

Hausmann’s Theorem

Convexity Radius

  • the largest (sup) radius so that geodesic balls are convex.

    • for the circle, it is a quarter of the circumference

    • denoted by \(\rho(M)\)

    • \(\rho(M)>0\) for a compact manifold

Theorem (Hausmann 1995): For any \(0<\beta<\rho(M)\), the Vietoris–Rips complex \(\mathcal{R}_\beta(M)\) is homotopy equivalent to \(M\).

  • vertex set is the entire manifold \(M\)
  • Question: what about the Vietoris–Rips complex of a dense subset \(X\subset M\)?

Latschev’s Theorem

(Latschev 2001)

Qualitative Latschev’s Theorem

Jungs’ Theorem

Classical

Manifolds

Discussions

  • How to retain all the metric properties for the GMD?
  • Can the GMD be useful in supervised learning for classification?
  • Currently, NONE of the graph distances (GED, GGD, GMD) is stable with respect another. Can we develop a stable graph distance?

References

Hausmann, Jean-Claude. 1995. “On the Vietoris-Rips Complexes and a Cohomology Theory for Metric Spaces.” In Prospects in Topology (AM-138), edited by Frank Quinn, 175–88. Proceedings of a Conference in Honor of William Browder. (AM-138). Princeton University Press.
Latschev, J. 2001. “Vietoris-Rips Complexes of Metric Spaces Near a Closed Riemannian Manifold.” Archiv Der Mathematik 77 (6): 522–28. https://doi.org/10.1007/PL00000526.